Question

Let the incentre of $\Delta ABC$ is $I\left(2,5\right)$ . If $A=\left(1,13\right)$ and $B=\left(-4,1\right)$ , then the coordinates of $C$ are

• A. $\left(1,10\right)$
• B. $\left(10,1\right)$
• C. $\left(8,2\right)$
• D. $\left(9,3\right)$

Answer 1

Answer: (B)

Slope of $AB$ is $\frac{13-1}{1-\left(-4\right)}=\frac{12}{5}$
Slope of $BI$ is $\frac{5-1}{2-\left(-4\right)}=\frac{2}{3}$
Slope of $AI$ is $\frac{13-5}{1-\left(2\right)}=-8$
Let, the slopes of $BC$ and $AC$ are $m_1$ & $m_2$ respectively
$\Rightarrow \frac{m_2-\left(-8\right)}{1+m_2\left(-8\right)}=\frac{-8-\left(\frac{12}{5}\right)}{1+\left(-8\right)\left(\frac{12}{5}\right)}$
$\left(m_2+8\right)\left(91\right)=\left(1-8m_2\right)\left(-57\right)$
$8\times 91-57=\left(8\times 57-91\right)m_2\Rightarrow m_2=-\frac{4}{3}$
Now, $\frac{m_1-\frac{2}{3}}{1+m_1\left(\frac{2}{3}\right)}=\frac{\frac{2}{3}-\frac{12}{5}}{1+\frac{2}{3}\times \frac{12}{5}}$
$\Rightarrow \left(3m_1-2\right)\left(39\right)=\left(3+2m_1\right)\left(-26\right)$
$\Rightarrow 9m_1-6=-6-4m_1\Rightarrow m_1=0$
$\Rightarrow$ The equation of $AC$ is $\left(y-13\right)=-\frac{4}{3}\left(x-1\right)$ and the equation of $BC$ is $y=1$
$\Rightarrow C$ is $\left(10,1\right)$